Question

Prove the identity 2csc2x=csc^2xtanz

Answer 1

Step-by-step explanation:

Consider the LHS, after the 5th step, consider the RHS

$2 \csc(2x) = \csc {}^{2} (x) \tan(x)$

$2 \frac{1}{ \sin(2x) } = \csc {}^{2} (x) \tan(x)$

$2 \times \frac{1}{2 \sin(x) \cos(x) } = \csc {}^{2} (x) \tan(x)$

$\frac{1}{ \sin(x) \cos(x) } = \csc {}^{2} (x) \tan(x)$

$\csc(x) \sec(x) = \csc {}^{2} (x) \tan(x)$

Consider the RHS

$\csc(x) \sec(x) =( 1 + \cot {}^{2} (x) ( \tan(x))$

$\csc(x) \sec(x) = \tan(x) + \cot(x)$

$\csc(x) \sec(x) = \frac{ \sin(x) }{ \cos(x) } + \frac{ \cos(x) }{ \sin(x) }$

$\csc(x) \sec(x) = \frac{1}{ \sin(x) \cos(x) }$

$\csc(x) \sec(x) = \csc(x) \sec(x)$